Paths in Polynomial Lemniscates: A Degree-Seven Counterexample and Radial Connections

Will Cook

Long paper

Paths in Polynomial Lemniscates: A Degree-Seven Counterexample and Radial Connections

Erdős #1041 By Will Cook 29 September 2026 Authorship, AI use and citation

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Summary. The record gives the rational estimates and geometric proof for ani's degree-seven polynomial: every connected set in its strict unit lemniscate joining two roots has Hausdorff measure greater than two. It develops contained paths for trinomials, collinear roots and two sparse families. Ordinary analytic arguments for critical-value criteria retain their unformalised inputs beside the statements. Critical-value moments, counterexamples to prescribed paths and questions about flow-strip geometry complete the account. The one-polynomial formal statement and correspondence with the historical wording remain distinct.

This paper covers the exact geometry of ani's one-polynomial counterexample, proved positive families and critical-value arguments with their unformalised inputs identified; formal-statement status and historical correspondence remain owned by the claims registry.

It does not establish the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or the current status of Erdős #1041, including the later one-polynomial Formal Conjectures path-image Hausdorff-length check and its unreviewed historical curve-length correspondence; those distinctions belong to docs/claims.json in the mathematics repository.

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In this paper

Critical points and local estimates

The critical points cluster at scale $ε$ . We resolve them by writing $F (ε w) = - 1 + ε^{7} Q (w)$ , where

Q (w) = c_{0} w + b w^{2} + a w^{3} - ε \bar{a} w^{4} - ε^{3} \bar{b} w^{5} - ε^{5} {\bar{c}}_{0} w^{6} + w^{7} .

For a centre $v_{j}$ , let $T_{k} (v_{j})$ be the coefficient of $z^{k}$ in $Q (v_{j} + z)$ . If $Q (w) = \sum q_{k} w^{k}$ , these coefficients are given by the finite formula

T_{k} (v_{j}) = \sum_{ℓ = k}^{7} (\binom{ℓ}{k}) q_{ℓ} v_{j}^{ℓ - k} .

Use the following centres, whose displayed decimals are exact, and radius $r = 10^{- 6}$ :

$j$	$Re v_{j}$	$Im v_{j}$
0	$3.1898275786$	$- 0.4999999979$
1	$- 3.1898275845$	$- 0.5000000021$
2	$0.0000000039$	$0.8232474662$
3	$0.0000002274$	$1.8078590858$
4	$- 0.0000002265$	$1.8838591455$
5	$0.0000000011$	$- 3.5149656976$

These six discs are disjoint. For a Gaussian rational $z = u + i v$ , put $N (z) = | u | + | v |$ and $m (z) = max (| u |, | v |)$ , so $m (z) \leq | z | \leq N (z)$ . Define

E_{j} = N (T_{1} (v_{j})) + \sum_{k = 2}^{6} (k + 1) N (T_{k + 1} (v_{j})) r^{k}, L_{j} = 2 m (T_{2} (v_{j})) r .

The next table proves $E_{j} < L_{j}$ in every disc. Rouché’s theorem, applied to the linear term of $Q^{'}$ , gives exactly one critical point in each.

To determine its critical-value modulus, use

H (w) = \frac{1 - ρ^{14}}{2 ρ^{14} ε^{7}} + Re Q (w) - \frac{ε^{7}}{2} | Q (w) |^{2}, | f (ρ ε w) |^{2} = 1 - 2 ρ^{14} ε^{7} H (w) .

For $| w - v_{j} | \leq r$ , set $V_{j} = \sum_{k = 1}^{7} N (T_{k} (v_{j})) r^{k}$ . Substituting $Re T_{0} (v_{j}) \pm V_{j}$ for $Re Q (w)$ and $N (T_{0} (v_{j})) + V_{j}$ for $| Q (w) |$ gives the following rational bounds. An interval in the last column contains $10^{9} H (w)$ throughout the whole disc.

$j$	$10^{9} E_{j} <$	$10^{6} L_{j} >$	$10^{9} H (w)$
0	423	8340	$[- 502, - 486]$
1	433	8340	$[- 502, - 486]$
2	12	378	$[3556, 3558]$
3	3	43	$[- 75, - 74]$
4	4	48	$[- 864, - 863]$
5	1298	16812	$[- 1566, - 1548]$

It follows that $f$ has just one critical point in ${| f | < 1}$ , namely $c_{s} = ρ ε w_{s}$ with $| w_{s} - v_{2} | < r$ . Its critical value is nonzero, since the polynomial has simple roots, and no critical point lies on $| f | = 1$ .

Expanding the same Taylor coefficients throughout $| w - v_{2} | \leq r$ gives

| T_{2} (w) | > 189.1, (| T_{3} (w) |, | T_{4} (w) |, | T_{5} (w) |, | T_{6} (w) |, | T_{7} (w) |) < (190, 20, 15, 6, 2) .

In the last coordinate the exact value is $| T_{7} | = 1$ . In particular, for $h = ρ ε / 10$ and $f (c_{s} + z) - f (c_{s}) = z^{2} A_{s} (z)$ ,

\begin{matrix} (3) & | A_{s} (0) | > 180 ρ^{5} ε^{5}, | \frac{A_{s} (z)}{A_{s} (0)} - 1 | < \frac{1}{4} (| z | \leq h) . \end{matrix}

Indeed the numerator of the relative error is at most $190 / 10 + 20 / 100 + 15 / 1000 + 6 / 10^{4} + 1 / 10^{5} < 189.1 / 4$ in scaled coordinates. The positive critical-value bound in the table also gives

0 < δ = 1 - | f (c_{s}) | \leq 1 - | f (c_{s}) |^{2} < ρ^{7} ε^{7} \frac{36}{5 \cdot 10^{6}} .

Consequently

\begin{matrix} (4) & \sqrt{δ / | A_{s} (0) |} < \frac{ρ ε}{5000}, δ < | A_{s} (0) | h^{2} / 4. \end{matrix}

About this paper

Authorship and AI use. Will Cook built and directed the research infrastructure and maintains the public release. He reviewed claims when he could. AI agents did most of the research and drafting. Cook did not independently verify every claim.

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